Conformal Maps and p-Dirichlet Energies

Let Φ : (M1, g1)→ (M2, g2) be a diffeomorphism between Riemannian manifolds and Φ# : D(M2)→ D(M1) the induced pull-back operator. The main theorem of this work is Theorem 4.1 which relates preservation of the p-Dirichlet energies φ 7→ ∫ Mi |dφ| dμi under Φ# to isometric or conformal properties of Φ. More precisely: In case p = n, Φ# preserves the p-Dirichlet energy if and only if Φ is conformal. In case n 6 = p ≥ 2, energy preservation is equivalent to Φ being an isometry.